WHAT YOU KNOW: We used the rectangular coordinate system to represent ordered pairs of real numbers and to graph equations in two variables. We saw that linear equations can be written in the form a x + b = 0 , a ≠ 0 , and quadratic equations can be written in the general form a x 2 + b x + c = 0 , a ≠ 0 . We solved linear equations. We saw that some equations have no solution, whereas others have all real numbers as solutions. We solved quadratic equations using factoring, the square root property, completing the square, and the quadratic formula. We saw that the discriminant of a x 2 + b x + c = 0 , b 2 − 4 a c , determines the number and type of solutions. We performed operations with complex numbers and used the imaginary unit i ( i = − 1 , where i 2 = − 1 ) to represent solutions of quadratic equations with negative discriminants. Only real solutions correspond to x -intercepts. We also solved rational equations by multiplying both sides by the least common denominator and clearing fractions. We developed a strategy for solving a variety of applied problems, using equations to model verbal conditions. In Exercises 1-12, solve each equation. 4 x − 2 ( 1 − x ) = 3 ( 2 x + 1 ) − 5
WHAT YOU KNOW: We used the rectangular coordinate system to represent ordered pairs of real numbers and to graph equations in two variables. We saw that linear equations can be written in the form a x + b = 0 , a ≠ 0 , and quadratic equations can be written in the general form a x 2 + b x + c = 0 , a ≠ 0 . We solved linear equations. We saw that some equations have no solution, whereas others have all real numbers as solutions. We solved quadratic equations using factoring, the square root property, completing the square, and the quadratic formula. We saw that the discriminant of a x 2 + b x + c = 0 , b 2 − 4 a c , determines the number and type of solutions. We performed operations with complex numbers and used the imaginary unit i ( i = − 1 , where i 2 = − 1 ) to represent solutions of quadratic equations with negative discriminants. Only real solutions correspond to x -intercepts. We also solved rational equations by multiplying both sides by the least common denominator and clearing fractions. We developed a strategy for solving a variety of applied problems, using equations to model verbal conditions. In Exercises 1-12, solve each equation. 4 x − 2 ( 1 − x ) = 3 ( 2 x + 1 ) − 5
Solution Summary: The author calculates the solution set of the given equation as 4x-2(1-x)=3 (2x+1 )-5.
WHAT YOU KNOW: We used the rectangular coordinate system to represent ordered pairs of real numbers and to graph equations in two variables. We saw that linear equations can be written in the form
a
x
+
b
=
0
,
a
≠
0
, and quadratic equations can be written in the general form
a
x
2
+
b
x
+
c
=
0
,
a
≠
0
. We solved linear equations. We saw that some equations have no solution, whereas others have all real numbers as solutions. We solved quadratic equations using factoring, the square root property, completing the square, and the quadratic formula. We saw that the discriminant of
a
x
2
+
b
x
+
c
=
0
,
b
2
−
4
a
c
, determines the number and type of solutions. We performed operations with complex numbers and used the imaginary unit
i
(
i
=
−
1
,
where
i
2
=
−
1
)
to represent solutions of quadratic equations with negative discriminants. Only real solutions correspond to x-intercepts. We also solved rational equations by multiplying both sides by the least common denominator and clearing fractions. We developed a strategy for solving a variety of applied problems, using equations to model verbal conditions.
In Exercises 1-12, solve each equation.
4
x
−
2
(
1
−
x
)
=
3
(
2
x
+
1
)
−
5
Formula Formula A polynomial with degree 2 is called a quadratic polynomial. A quadratic equation can be simplified to the standard form: ax² + bx + c = 0 Where, a ≠ 0. A, b, c are coefficients. c is also called "constant". 'x' is the unknown quantity
You are given a plane Π in R3 defined by two vectors, p1 and p2, and a subspace W in R3 spanned by twovectors, w1 and w2. Your task is to project the plane Π onto the subspace W.First, answer the question of what the projection matrix is that projects onto the subspace W and how toapply it to find the desired projection. Second, approach the task in a different way by using the Gram-Schmidtmethod to find an orthonormal basis for subspace W, before then using the resulting basis vectors for theprojection. Last, compare the results obtained from both methods
Plane II is spanned by the vectors:
- (2) · P² - (4)
P1=2
P21
3
Subspace W is spanned by the vectors:
2
W1
- (9) ·
1
W2
1
= (³)
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.
Linear Equation | Solving Linear Equations | What is Linear Equation in one variable ?; Author: Najam Academy;https://www.youtube.com/watch?v=tHm3X_Ta_iE;License: Standard YouTube License, CC-BY